Question

A man starts at the origin O and walks a distance of 3 units in the north-east direction and then walks a distance of 4 units in the north-west direction to reach the point P. Then \( \overline{OP} \) is equal to:

• A. \( \frac{1}{\sqrt{2}}(-\hat{i}+\hat{j}) \)
• B. \( \frac{1}{2}(\hat{i}+\hat{j}) \)
• C. \( \frac{1}{\sqrt{2}}(\hat{i}-7\hat{j}) \)
• D. \( \frac{1}{\sqrt{2}}(-\hat{i}+7\hat{j}) \)

Hint:

Remember the trigonometric values for standard angles like \( 45^{\circ} \) and \( 135^{\circ} \).

Answer 1

The Correct Option is D

Step 1: Represent the first displacement vector \( \vec{D_1} \) (north-east). \[ \vec{D_1} = 3 \cos(45^{\circ}) \hat{i} + 3 \sin(45^{\circ}) \hat{j} = \frac{3}{\sqrt{2}} \hat{i} + \frac{3}{\sqrt{2}} \hat{j} \] Step 2: Represent the second displacement vector \( \vec{D_2} \) (north-west). \[ \vec{D_2} = 4 \cos(135^{\circ}) \hat{i} + 4 \sin(135^{\circ}) \hat{j} = -\frac{4}{\sqrt{2}} \hat{i} + \frac{4}{\sqrt{2}} \hat{j} \] Step 3: Find the resultant displacement vector \( \overline{OP} = \vec{D_1} + \vec{D_2} \). \[ \overline{OP} = \left(\frac{3}{\sqrt{2}} - \frac{4}{\sqrt{2}}\right) \hat{i} + \left(\frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}}\right) \hat{j} \] Step 4: Simplify the components. \[ \overline{OP} = -\frac{1}{\sqrt{2}} \hat{i} + \frac{7}{\sqrt{2}} \hat{j} = \frac{1}{\sqrt{2}}(-\hat{i} + 7\hat{j}) \]